What do "T" and " " mean? If T is the variable temperature and is a constant temperature, then means temperature T is greater than the the temperature and T will decrease to . So dT/dt must be negative and that means that k must be negative.
By the way, although it wasn't asked, if T< , so T increases to . dT/dt is positive now which means that, since T- is negative, k must still be negative. The "k" in Newton's law of cooling is always negative.

Find the GS of the DE . You may assume and the final solution should give explicitly in terms of and a constant of integration.

Could someone give me a helping hand with this? I'm new to differential equations and this question was just thrown at me unexpectedly.

"Separate" it: . Integrate both sides. You will get a logarithm on the left so solving for T will give an exponential of kt.